The maximum output voltage of a 3-phase voltage source inverter (VSI) is obtained when it is operated in six step mode. In this mode and referring to FIG. 1, a reference voltage vector V* occupies six discrete positions V1, V2, V3, V4, V5, and V6. In this mode, the peak value of the fundamental phase voltage is 2VDC/π, where VDC is the value of the DC bus voltage at the inverter input. A figure of merit, or modulation index, can be defined as:                               m          =                                    v                              p                ⁢                                                                  ⁢                k                                                    2              ⁢                                                V                                      D                    ⁢                                                                                  ⁢                    C                                                  /                π                                                    ,                            (        1        )            where vpk is the peak value of the fundamental of the output phase voltage of the inverter. The modulation index m is introduced for the purpose of characterizing other modes of the inverter operation.
When the vector V* follows a circle 10 inscribed in hexagon 12 defined by discrete positions V1, V2, V3, V4, V5, and V6, the value of the modulation index is m=0.785. The use of space vector modulation, where reference vector V* follows the perimeter of hexagon 12 increases the modulation index to 0.908. The operating region between m=0.908 and m=1 is defined as “overmodulation.” The use of overmodulation in VSI motor drives improves DC bus voltage utilization and maximizes the AC output voltage of the inverter. These improvements can be useful in VSI powered motor drives, where maximum motor torque is proportional to the square of the applied AC voltage.
Although overmodulation increases the output fundamental voltage of the inverter by about 10%, it also introduces superior harmonics. It is known that the line-to-neutral output voltage Vl-n of an inverter in six-step mode can be written:                               V                      l            -            n                          =                                                            2                ⁢                                  V                                      d                    ⁢                                                                                  ⁢                    c                                                              π                        ⁡                          [                                                sin                  ⁡                                      (                                          ω                      ⁢                                                                                          ⁢                      t                                        )                                                  -                                                      1                    5                                    ⁢                                      sin                    ⁡                                          (                                              5                        ⁢                        ω                        ⁢                                                                                                  ⁢                        t                                            )                                                                      -                                                      1                    7                                    ⁢                                      sin                    ⁡                                          (                                              7                        ⁢                        ω                        ⁢                                                                                                  ⁢                        t                                            )                                                                      -                …                            ]                                .                                    (        2            
The 6n±1 order harmonics in the line-to-neutral voltages are typical for a Y-connected load, which is the case of electric motors. The harmonic voltages will generate harmonic currents of the same order into the motor supplied by the inverter. This situation is illustrated by FIG. 2, in which waveform 14 represents stator flux position angle, waveform 18 represents stator phase A current, and waveform 16 represents a stator current spectrum. The 5th harmonic (630 Hz=126 Hz×5) and the 7th harmonic (882 Hz=126 Hz×7) of the phase current of an induction motor operated in the overmodulation region can be observed.
To apply field-oriented control of the motor, the motor phase currents are passed through a coordinate transformation, from the stationary reference frame to the rotor field reference frame. Consequently, the 6n±1 order harmonics become ±6n harmonics and are superimposed on the stationary reference frame stator currents, which are DC values. This superimposing can be observed in FIG. 3, which is an oscillogram recorded under conditions similar to those of the oscillogram of FIG. 2. However, in FIG. 3, the spectrum of the q-axis current is shown. More particularly, waveform 20 represents stator flux position angle, waveform 24 represents q-axis current, and waveform 22 represents a q-axis current spectrum. The dominant harmonics are the 6th and the 12th, which are at approximately 750 Hz and 1.5 kHz. Similar ripple will appear in the d-axis current. The effect on the rotor flux, and thus, on the motor torque, is greatly reduced, because the ripple is filtered by the rotor circuit with the rotor time constant, if the motor is an induction motor.
The q-axis current 24 depicted by the oscillogram in FIG. 3 is applied as negative feedback at the input of a current controller of a field-oriented motor drive system. If a high bandwidth current controller is used for achieving good dynamic performance of the motor torque, the ±6n harmonic ripple can pass through the current controller. The most significant components of this ripple are the 6th and 12th harmonics, which can pass through the current controller and can render the operation of the motor drive unstable, even in steady state. This problem is evident in the oscillograms of FIG. 4 and FIG. 5, in which the phase current waveforms 26, 28, 30, and 32 are irregular and exhibit increased ripple. (In FIG. 4, stator flux position angle is represented by waveform 34 and q-axis current by waveform 36.)
One method known to reduce this instability is to lower the gain of the current controller, which in turn lowers its bandwidth. The decreased bandwidth avoids perturbation of the system by high order harmonics present in the current feedback. However, this solution trades tight current regulation for stability, and tight current regulation is useful in high dynamic response motor drive systems or motor drives operating over a wide field weakening range.
In Holtz et al., “On Continuous Control of PWM Inverters in the Overmodulation Range Including Six-Step Mode,” Proc. IECON '92, pp. 307–312, 1992, two overmodulation modes are described. In one mode, a microprocessor changes the magnitude of a reference without changing its angle. Pulsewidth control in this overmodulation range can be carried out as long as a portion of the track of V* exists within hexagon 12. In a second overmodulation mode, modulation index is higher than in the first mode, and the processor changes both the magnitude and phase of the reference. Current distortion can be decreased by increasing the switching frequency. Errors in feedback due to overmodulation are discussed, and a switching algorithm is proposed that selects a switching state providing a negative rate of change of current in the d-direction. Compensating for selected harmonic components is not shown or suggested.
In Khambadkone et al., “Compensated Synchronous PI Current Controller in Overmodulation Range and Six-Step Operation of Space-Vector-Modulation-Based Vector Controlled Drives, IEEE Trans. on Indust. Elect., vol. 49, pp. 574–580, June 2002, a harmonic estimation correction to feedback is proposed for a controller. Current control is carried out in field coordinates, and harmonic disturbances are compensated before a feedback current signal is fed to a controller. A total harmonic voltage correction is estimated to perform this correction. The total harmonic voltage correction is estimated utilizing a first order approximation in field coordinates. The harmonic current error due to overmodulation is compensated by the estimated value. Compensating for selected harmonic components is not shown or suggested.